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Mashups Using Mathematical Knowledge

Why Formulas Are Different

  • Chapter
Semantic Mashups

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Abstract

Mashups offer new functionality by combining, aggregating and transforming resources and services available on the Web. This chapter deals with mathematical mashups and focuses on those that process mathematical knowledge rather than, e.g., huge amounts of numeric data, as it is the structure of knowledge that distinguishes mathematics from other application domains.

The resources that mathematical mashups process primarily include formulas and the services they offer involve computation. In knowledge-rich mashups, the formulas are not hard-coded into the implementation but represented as explicit data structures, and often also presented to the user. These structures are different from the (re)presentations mashups usually process. To allow for automated processing, formulas need to be represented neither as plain text nor as images, but in a symbolic way. The representation of choice is, for compatibility and scalability reasons discussed in this chapter, usually not JSON or RDF, but semantic XML markup. Besides tables or graphs, mathematical mashups may also require formulas to be presented to the end user. The highest degree of interaction with formulas is offered by MathML—in those browsers that fully support it.

After introducing typical education and engineering use cases that benefit from mathematical mashups, this chapter reviews the conceptual and technical foundations for representing and presenting mathematical formulas, discussing MathML as well as alternatives. We continue with a review of mathematical web services and collections of mathematical knowledge that provide suitable building blocks for mathematical mashups. We then present the Planetary system, a math-enabled social semantic web portal that provides an environment for executable papers, and the SAlly framework that mashes up user interfaces of software applications with mathematical web services. Both environments mash up assistive services by hooking them into document structures, which have been annotated with terms from a mathematical background ontology. We conclude with an outlook towards contributing collections of mathematical knowledge to the Web of Data, and outline how such linked open datasets can drive further mathematical mashups in the near future.

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Notes

  1. 1.

    This notion of the term “knowledge management” is wider than that of its traditional definition as “a range of practices used […] to identify, create, represent, distribute and enable adoption of insights and experiences. Such insights and experiences comprise knowledge, either embodied in individuals or embedded in organizational processes or practice” [66].

  2. 2.

    Numbering added by the authors.

  3. 3.

    To reduce eye strain, we only capitalize this term, as well as the terms “Web 2.0” and “Semantic Web”, when they denote the Web as a whole, but not when they are in an adjective position, as in “semantic web services”.

  4. 4.

    Other than the above-mentioned wiki front-end to the Mizar Mathematical Library and a similar one for the library of the Coq theorem prover, none of which are the primary entry points to these libraries yet, we are not aware of mathematical web 2.0 sites that integrate formal verification.

  5. 5.

    Similarly, Baez suggests that the release of a formula editor plugin for the popular WordPress blog engine was a major incentive for mathematicians to start blogging [13].

  6. 6.

    But see Donald Byrd’s documentation of “extremes of conventional music notation” [19].

  7. 7.

    The information on browser support has been taken from Wikipedia [91], “When can I use…” [134], and a review by Timothy Vismor [131].

  8. 8.

    If we do not have the information that this is a single formula, then additional readings appear.

  9. 9.

    Even in Presentation MathML, it is, however, best practice to mark up the distinction between multiplication and function application explicit using the special Unicode characters Function Application (U+2061) and Invisible Times (U+2062). Both characters occupy no space on the screen and are thus invisible.

  10. 10.

    “An approach to analysing the usability of information artifacts” [16], which has also been applied to mashups [40, 80].

  11. 11.

    The original abbreviation means “JavaScript API for OMDoc-Based Interactive Documents”, but JOBAD-enabled documents do not have to be generated from an OMDoc source.

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Acknowledgements

The authors would like to thank Paul Libbrecht for his thorough review and constructive feedback.

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Authors and Affiliations

  1. School of Computer Science, University of Birmingham, Birmingham, UK

    Christoph Lange

  2. Computer Science, Jacobs University Bremen, Bremen, Germany

    Michael Kohlhase

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  1. Christoph Lange
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  2. Michael Kohlhase
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Correspondence to Christoph Lange .

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    Brigitte Endres-Niggemeyer

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Lange, C., Kohlhase, M. (2013). Mashups Using Mathematical Knowledge. In: Endres-Niggemeyer, B. (eds) Semantic Mashups. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36403-7_6

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  • DOI: https://doi.org/10.1007/978-3-642-36403-7_6

  • Publisher Name: Springer, Berlin, Heidelberg

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